Question

prove all the theorams of circle ​

Answer 1

Step-by-step explanation:

statements proving the theorem into the correct order.

The angle at the centre of a circle is twice the angle at the circumference

Circle Theorem

Similarly ∠AOC = 180° – 2 x ∠OCA

OB = OC (radii of circle)

∠BOA = 2∠BCA Q.E.D.

Construct radius OC

∠COB = 180° – 2 x ∠BCO (Angle sum of triangle OBC)

To prove: ∠BOA = 2∠BCA

∠BCO = ∠OBC (equal angles in isosceles triangle)

∠BOA = 2(∠BCO + ∠OCA)

∠BOA = 360° – (180° – 2 x ∠BCO + 180° – 2 x ∠OCA)

∴ OBC is an isosceles triangle

The angles in the same segment are equal

Circle Theorem

∴ 2 x ∠ABD = 2 x ∠ACD

∠AOD = 2 x ∠ABD (angle at centre twice angle at circumference)

Construct radii from A and D

∠AOD = 2 x ∠ACD (angle at centre twice angle at circumference)

∠ABD = ∠ACD Q.E.D.

To prove: ∠ABD = ∠ACD

The angle in a semi-circle is a right angle

Circle Theorem

Similarly in triangle BCO ∠OCB = ∠OBC

To prove: ∠ABC = 90°

∴ ∠OAB = ∠OBA (equal angles in isosceles triangle ABO)

∠OAB + ∠OBA + ∠OCB + ∠OBC = 180° (Angle sum of triangle ABC)

OA = OB (radii)

∴ 2(∠OBA + ∠OBC) = 180°

∴ ABO is an isosceles triangle (two equal sides)

∠ABC = 90° Q.E.D.

Construct the radius OB

∴ ∠OBA + ∠OBC = 90°

Opposite angles of a cyclic quadrilateral are supplementary

Circle Theorem

The obtuse and reflex angles at O add up to 360° (angles at a point)

Similarly the obtuse angle AOC = 2 x ∠CDA

To prove ∠ABC + ∠CDA = 180°

∴ 2 x ∠ABC + 2 x ∠CDA = 360°

Reflex ∠AOC = 2 x ∠ABC (angle at centre twice angle at circumference)

∠ABC + ∠CDA = 180° Q.E.D.

Construct the radii OA and OC

The angle between the tangent and a chord is equal to the angle in the alternate segment.

Circle Theorem

2 x ∠CAB = 2 x ∠CBD (from [1] above)

∠OBC + ∠CBD = 90° (angle between radius and tangent) [2]

∠CAB = ∠CBD Q.E.D.

2 x ∠OBC + ∠COB = 180° (angle sum of triangle) [3]

Obtuse ∠COB = 2 x ∠CAB (angle at centre twice angle at circumference) [1]

∠COB = 2 x ∠CBD

To prove ∠CAB = ∠CBD

∠OBC = ∠OCB (equal angles in isosceles triangle OBC)

Construct the radii OB and OC

2 x ∠OBC + ∠COB = 2

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Hope it helps you.

Answer 2

Hi mate

here's your answer

• The angle subtended at the centre of a circle is double the size of the angle subtended at the edge from the same two points,
• Angles which are in the same segment are equal, i.e. angles subtended (made) by the same arc at the circumference are equal,
• The angles which are within a semicircle add up to 90°,
• Opposite angles in a cyclic quadrilateral add up to 180°,
• Alternate Segment Theorem, i.e. that the angle between a tangent and its chord is equal to the angle in the 'alternate segment'.

hope this helps you

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